Finite groups with some bounded codegrees

For a character $\chi$ of a finite group $G$, the number cod$(\chi ):=\mathrm{\mid }G:\mathrm{k}\mathrm{e}\mathrm{r}(\chi )\mathrm{\mid }\mathrm{/}\chi (1)$ is called the codegree of $\chi$.In this paper, we give a solvability criterion for a finite group $G$ depending on the minimum of the ratio $\chi (1{)}^2\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{d}(\chi )$, when $\chi$ varies among the irreducible characters of $G$.